> For the complete documentation index, see [llms.txt](https://algorithm.prettylog.com/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://algorithm.prettylog.com/top-75-leetcode-questions-to-save-your-time/problems/dynamic-programming/x-62.-unique-paths.md).

# x 62. Unique Paths

{% embed url="<https://leetcode.com/problems/unique-paths/description/>" %}

### Recursive

```javascript
 const memo = {};
 
 const uniquePaths = function(m, n) {
     const key = m + ',' + n;
     if (key in memo) return memo[key];
     if (m === 1 || n === 1) return 1
     if (m === 0 || n === 0) return 0;
     
     memo[key] = uniquePaths(m-1, n) + uniquePaths(m, n-1);
     return memo[key];
};
```

###

### 2nd try

> time: O(m x n)

> space: O(m or n)

```jsx

// this reduces time
// const memo = {};
const memo = {};
var uniquePaths = function (m, n) {

    if (memo['' + m + n] !== undefined) {
        return memo['' + m + n];
    }

    const arr = new Array(m).fill(1);
    for (let i = 1; i < n; i++) {
        for (let j = 1; j < m; j++) {
            arr[j] += arr[j - 1];
        }
    }

    memo['' + m + n] = arr[m - 1];
    return arr[m - 1];
};
```

### 1st try

> time: O(m x n)

> space: O(m x n)

```jsx
/**
 * @param {number} m
 * @param {number} n
 * @return {number}
 */
var uniquePaths = function(m, n) {
    const matrix = new Array(m);

    matrix[0] = new Array(n).fill(1);
    for (let i = 1; i < m; i++) {
        matrix[i] = new Array(n);
        matrix[i][0] = 1;
    }

    for (let i = 1; i < m; i++) {
        for (let j = 1; j < n; j++) {
            matrix[i][j] = matrix[i - 1][j] + matrix[i][j - 1];
        }
    }

    return matrix[m - 1][n - 1];
};
```


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